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The problem of finding a maximum size matching in a graph (known as the maximum matching problem) is one of the most classical problems in computer science. Despite a significant body of work dedicated to the study of this problem in the data stream model, the state-of-the-art single-pass semi-streaming algorithm for it is still a simple greedy algorithm that computes a maximal matching, and this way obtains 1 / 2 -approximation. Some previous works described two/three-pass algorithms that improve over this approximation ratio by using their second and third passes to improve the above mentioned maximal matching. One contribution of this paper continues this line of work by presenting new three-pass semi-streaming algorithms that work along these lines and obtain improved approximation ratios of 0.6111 and 0.5694 for triangle-free and general graphs, respectively. Unfortunately, a recent work Konrad and Naidu (Approximation, randomization, and combinatorial optimization. Algorithms and techniques, APPROX/RANDOM 2021, August 16–18, 2021. LIPIcs, vol 207, pp 19:1–19:18, 2021. https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2021.19) shows that the strategy of constructing a maximal matching in the first pass and then improving it in further passes has limitations. Additionally, this technique is unlikely to get us closer to single-pass semi-streaming algorithms obtaining a better than 1 / 2 -approximation. Therefore, it is interesting to come up with algorithms that do something else with their first pass (we term such algorithms non-maximal-matching-first algorithms). No such algorithms were previously known, and the main contribution of this paper is describing such algorithms that obtain approximation ratios of 0.5384 and 0.5555 in two and three passes, respectively, for general graphs. The main significance of our results is not in the numerical improvements, but in demonstrating the potential of non-maximal-matching-first algorithms. [ABSTRACT FROM AUTHOR]

In a recent paper, Avis, Bremner, Tiwary and Watanabe gave a method for constructing linear programs (LPs) based on algorithms written in a simple programming language called Sparks. If an algorithm produces the solution x to a problem in polynomial time and space then the LP constructed is also of polynomial size and its optimum solution contains x as well as a complete execution trace of the algorithm. Their method led us to the construction of a compiler called sparktope which we describe in this paper. This compiler allows one to generate polynomial sized LPs for problems in P that have exponential extension complexity, such as matching problems in non-bipartite graphs. In this paper, we describe sparktope, the language Sparks, and the assembler instructions and LP constraints it produces. This is followed by two concrete examples, the makespan problem and the problem of testing if a matching in a graph is maximum, both of which are known to have exponential extension complexity. Computational results are given. In discussing these examples, we make use of visualization techniques included in sparktope that may be of independent interest. The extremely large linear programs produced by the compiler appear to be quite challenging to solve using currently available software. Since the optimum LP solutions can be computed independently they may be useful as benchmarks. Further enhancements of the compiler and its application are also discussed. [ABSTRACT FROM AUTHOR]

A critical prerequisite for controlling complex networks is to find a driver node set with a structural controllability guarantee. This paper introduces the control capacity for a driver node set and solves the problem of finding a complete minimum driver node set that not only guarantees network structural controllability but achieves the desired level of control capacity. A novel algorithmic framework is proposed which is based on the concept of equivalent set and approximate matching replacement technique. The proposed algorithmic framework is shown to outperform the state-of-the-art approaches in the literature. The validity of the proposed algorithm is analyzed and the performance is evaluated by experiments on artificial and real-world complex networks. [ABSTRACT FROM AUTHOR]

Nowadays, large cities face numerous problems caused by the rapidly increasing number of vehicles on road. Carpooling has been shown as an efficient solution to ease the traffic pressure. The objective of our carpool scheduling problem is to minimise the number of carpools needed to transport users. Previous research on the similar topic introduces static capacity constraint which limits the carpool size to the vehicle's capacity. However, it is unnecessary since a seat in the vehicle can be occupied by several passengers if their routes are not overlapped. In this paper, we remove this constraint, which allows a vehicle to serve more users than its capacity. We propose a greedy approach based on the iterative matching and merging. Specifically, starting from a set of single-user carpools, the algorithm iteratively checks the merge-ability between carpools, and then applies the maximum matching algorithm to maximise the number of carpools merged. In addition, two methods are proposed to reduce the time complexity of merge-ability checking process. Furthermore, we improve the time efficiency of our algorithms by exploiting geometry properties. Experimental results on synthetic and real-world datasets show that our algorithms have better performances than baseline. [ABSTRACT FROM AUTHOR]

We study a natural generalization of the maximum weight many-to-one matching problem. We are given an undirected bipartite graph $$G= (A\, \dot{\cup }\, P, E)$$ with weights on the edges in E, and with lower and upper quotas on the vertices in P. We seek a maximum weight many-to-one matching satisfying two sets of constraints: vertices in A are incident to at most one matching edge, while vertices in P are either unmatched or they are incident to a number of matching edges between their lower and upper quota. This problem, which we call maximum weight many-to-one matching with lower and upper quotas (WMLQ), has applications to the assignment of students to projects within university courses, where there are constraints on the minimum and maximum numbers of students that must be assigned to each project. In this paper, we provide a comprehensive analysis of the complexity of WMLQ from the viewpoints of classical polynomial time algorithms, fixed-parameter tractability, as well as approximability. We draw the line between $$\textsf {NP}$$ -hard and polynomially tractable instances in terms of degree and quota constraints and provide efficient algorithms to solve the tractable ones. We further show that the problem can be solved in polynomial time for instances with bounded treewidth; however, the corresponding runtime is exponential in the treewidth with the maximum upper quota $$u_{\max }$$ as basis, and we prove that this dependence is necessary unless $$\textsf {FPT}= \textsf {W}[1]$$ . The approximability of WMLQ is also discussed: we present an approximation algorithm for the general case with performance guarantee $$u_{\max }+1$$ , which is asymptotically best possible unless $$\textsf {P}= \textsf {NP}$$ . Finally, we elaborate on how most of our positive results carry over to matchings in arbitrary graphs with lower quotas. [ABSTRACT FROM AUTHOR]

A connected graph $$G$$ is said to be factor-critical if $$G-v$$ has a perfect matching for every vertex $$v$$ of $$G$$ . Lovász proved that every factor-critical graph has an ear decomposition. In this paper, the ear decomposition of the factor-critical graphs $$G$$ satisfying that $$G-v$$ has a unique perfect matching for any vertex $$v$$ of $$G$$ with degree at least 3 is characterized. From this, the number of maximum matchings of factor-critical graphs with the special ear decomposition is obtained. [ABSTRACT FROM AUTHOR]

Controlling complex networks are one of the ultimate goals in network science. Previous works have found there exist two distinct control modes when controlling the dense networks: distributed and centralized modes. How to change the control mode of a network is a challenging problem. This paper presents an efficient algorithm to alter a network from distributed mode to centralized mode by reversing the direction of a few edges. We first analyze four possible cases when reversing an edge and then design an efficient algorithm to change the control mode of a network. We prove that our algorithm does not affect the control scheme of the network after mode change. We evaluate the performance of our algorithm on both synthetic and real networks. The results show that the control mode of most networks can be easily changed by reversing very few edges. Furthermore, the number of the possible driver nodes of the network after mode change is dramatically decreased, which means these networks are easier to control. Our algorithm provides the ability to design the desired control mode of a network for different control scenarios, which may be used in many applications. • We present a novel algorithm to change the control mode of a network by reversing the edges. • The control scheme remains unchanged when we change a network from distributed mode to centralized mode. • The results showed our algorithms are efficient and effective to change the control mode of complex networks. [ABSTRACT FROM AUTHOR]